What are indices?
An index (plural: indices), also called a power or exponent, tells you how many times a number is multiplied by itself. In 2⁵, the 2 is the base and the 5 is the index — it means 2 × 2 × 2 × 2 × 2. This connects directly to the Surds page, since roots are really fractional indices in disguise.
| Rule | In symbols |
|---|---|
| Multiplying powers of the same base | aᵐ × aⁿ = aᵐ⁺ⁿ |
| Dividing powers of the same base | aᵐ ÷ aⁿ = aᵐ⁻ⁿ |
| Power of a power | (aᵐ)ⁿ = aᵐⁿ |
| Zero index | a⁰ = 1 |
| Negative index | a⁻ⁿ = 1/aⁿ |
| Fractional index | a^(1/n) = ⁿ√a |
Multiplying and dividing powers
Worked Example 1
Simplify 3⁴ × 3²
1
Same base (3), so add the indices: 4 + 2 = 6
2
3⁶ = 729
Answer3⁶ = 729
Worked Example 2
Simplify 5⁷ ÷ 5³
1
Same base (5), so subtract the indices: 7 − 3 = 4
2
5⁴ = 625
Answer5⁴ = 625
Multiplying adds the indices; dividing subtracts them — but only when the base is the same.
Power of a power
Worked Example 3
Simplify (2³)⁴
1
A power raised to another power means multiplying the indices: 3 × 4 = 12
2
2¹² = 4096
Answer2¹² = 4096
Negative and fractional indices
A negative index means "reciprocal": a⁻ⁿ = 1/aⁿ. A fractional index means "root": a^(1/n) is the nth root of a, and a^(m/n) means take the nth root, then raise to the power m (or the other way round).
Worked Example 4
Evaluate (a) 4⁻² and (b) 27^(1/3) and (c) 8^(2/3)
1
(a) A negative index gives a reciprocal: 4⁻² = 1/4² = 1/16
2
(b) A power of 1/3 means "cube root": 27^(1/3) = ³√27 = 3
3
(c) A power of 2/3 means "cube root, then square": 8^(2/3) = (³√8)² = 2² = 4
Answer(a) 1/16 (b) 3 (c) 4
Practice questions
Work through each question before checking the answers.
Foundation (Grade 3–5)
Q1Simplify 2³ × 2⁴.Foundation
Q2Simplify 7⁶ ÷ 7².Foundation
Q3Simplify (3²)³.Foundation
Q4Evaluate 5⁰.Foundation
Q5Evaluate 2⁻³.Foundation
Higher (Grade 5–7)
Q6Simplify x⁵ × x⁻².Higher
Q7Evaluate 16^(1/2).Higher
Q8Evaluate 8^(1/3).Higher
Q9Evaluate 25^(3/2).Higher
Q10Simplify (2x³)².Higher
Higher — Hard (Grade 8–9)
Q11Evaluate 9^(−1/2).Grade 8–9
Q12Evaluate 32^(2/5).Grade 8–9
Q13Solve 2ˣ = 32.Grade 8–9
Q14Simplify (a³b²)² ÷ a⁴, giving your answer in index form.Grade 8–9
Q15Solve 3^(2x−1) = 27.Grade 8–9
Answers
Foundation (Q1–Q5)
Q12⁷ = 128
Q27⁴ = 2401
Q33⁶ = 729
Q41
Q51/8
Higher (Q6–Q10)
Q6x³
Q74
Q82
Q9125(√25 = 5, then 5³)
Q104x⁶
Higher — Hard (Q11–Q15)
Q111/3(√9 = 3, then reciprocal)
Q124(⁵√32 = 2, then 2²)
Q13x = 5(2⁵ = 32)
Q14a²b⁴((a³b²)² = a⁶b⁴, then ÷ a⁴)
Q15x = 2(27 = 3³, so 2x − 1 = 3)
Common mistakes
Common Mistake 1
Multiplying the base instead of adding the indices
3⁴ × 3² is not 9⁶. The base stays the same (3); only the indices are added: 3⁴⁺² = 3⁶.
Common Mistake 2
Thinking a negative index makes the answer negative
4⁻² does not equal −16. A negative index means "reciprocal", not "negative value": 4⁻² = 1/4² = 1/16, which is positive.
Common Mistake 3
Applying index laws to different bases
2³ × 3² cannot be simplified using the index laws, because the bases (2 and 3) are different. The addition/subtraction rules only work when the base is identical.
Common Mistake 4
Getting the fractional index the wrong way round
In a^(m/n), the denominator (n) is the root, and the numerator (m) is the power. Mixing these up — e.g. squaring first when you should be cube-rooting — gives the wrong answer.
Exam tips
💡 Exam Tip 1
Check the bases match before applying a law
Before adding or subtracting indices, always confirm the bases are identical. If they're not, the index laws don't apply directly — you may need to rewrite one base first.
💡 Exam Tip 2
Take the root first for fractional indices
For a^(m/n), it's almost always easier to find the nth root first, then raise to the power m — rooting a small number is far simpler than raising a large number to a power first.
💡 Exam Tip 3
Rewrite negative indices as fractions immediately
As soon as you see a negative index, rewrite it as "1 over" the positive version — this removes the risk of forgetting to take the reciprocal later in the calculation.
💡 Exam Tip 4
Use index laws to solve equations with unknown powers
For equations like 3^(2x−1) = 27, rewrite both sides with the same base first (27 = 3³), then simply equate the indices — this turns a tricky-looking equation into simple algebra.
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